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- 1. 1|Page When the Angles of a Triangle Don’t add up to 180 Degrees (by TARUN GEHLOT)1. IntroductionDo the angles of a triangle add up to 180 degrees or π radians? The answer issometimes yes, sometimes no. Is this an important question? Yes,because it leads to an understanding that there are different geometries based ondifferent axioms or rules of the game of geometry. Is it a meaningful question?Well no, at least not until we have agreed on the meaning of the words angleand triangle, not until we know the rules of the game. In this article we brieflydiscuss the underlying axioms and give a simple proof that the sum of the anglesof a triangle on the surface of a unit sphere is not equal to π but to π plus the areaof the triangle. We shall use the fact that the area of the surface of a unit sphereis 4π.2. The Big TheoremBefore we can say what a triangle is we need to agree on what we mean by pointsand lines. We are working on spherical geometry (literally geometry on thesurface of a sphere). In this geometry the space is the surface of the sphere; thepoints are points on that surface, and the line of shortest distance between twopoints is the great circle containing the two points. A great circle (like the Equator)cuts the sphere into two equal hemispheres. This geometry has obviousapplications to distances between places and air-routes on the Earth.Rotating sphere showing great circleThursday, January 17, 2013
- 2. 2|PageThe angle between two great circles at a point P is the Euclidean angle betweenthe directions of the circles (or strictly between the tangents to the circles at P).This presents no difficulty in navigation on the Earth because at any given pointwe think of the angle between two directions as if the Earth were flat at thatpoint.A lune is a part of the surface of the sphere bounded by two great circles whichmeet at antipodal points. We first consider the area of a lune and then introduceanother great circle that splits the lune into triangles.Rotating sphere showing 4 lunesThursday, January 17, 2013
- 3. 3|PageLemma.The area of a lune on a circle of unit radius is twice its angle, that is if the angle ofthe lune is A then its area is 2A. Two great circles intersecting at antipodal pointsP and P divide the sphere into 4 lunes. The area of the surface of a unit sphereis 4π.The areas of the lunes are proportional to their angles at P so the area of a lunewith angle A isA2π×4π=2AExercise 1.What are the areas of the other 3 lunes? Do your 4 areas add up to 4π?Thursday, January 17, 2013
- 4. 4|PageCheck your answers here .The sides of a triangle ABC are segments of three great circles which actually cutthe surface of the sphere into eight spherical triangles. Between the two greatcircles through the point A there are four angles. We label the angle insidetriangle ABC as angle A, and similarly the other angles of triangle ABC as angle Band angle C.Rotating sphere showing 8 trianglesExercise 2Rotating the sphere can you name the eight triangles and say whether any ofthem have the same area? Check your answers here .Theorem.Thursday, January 17, 2013
- 5. 5|PageConsider a spherical triangle ABC on the unit sphere with angles A, B and C. Thenthe area of triangle ABC isA + B + C - π.The diagram shows a view looking down on the hemisphere which has the linethrough AC as its boundary. The regions marked Area 1 and Area 3 are lunes withangles A and C respectively. Consider the lunes through B and B. Triangle ABC iscongruent to triangle ABC so the bow-tie shaped shaded area, marked Area 2,which is the sum of the areas of the triangles ABC and ABC, is equal to the areaof the lune with angle B, that is equal to 2B..So in the diagram we see the areas of three lunes and, using the lemma, theseare:Thursday, January 17, 2013
- 6. 6|PageArea 1 = 2AArea 2 = 2BArea 3 = 2CIn adding up these three areas we include the area of the triangle ABC threetimes. HenceArea 1 + Area 2 + Area 3 = Area of hemisphere +2(Area of triangle ABC)2A + 2B + 2C = 2 π + 2(Area of triangle ABC)Area of triangle ABC =A+B+C-π.3. Non-Euclidean GeometrySometimes revolutionary discoveries are nothing more than actually seeing whathas been under our noses all the time. This was the case over the discovery ofNon-Euclidean Geometry in the nineteenth century. For some 2000 years afterEuclid wrote his Elements in 325 BC people tried to prove the parallel postulateas a theorem in the geometry from the other axioms but always failed and that isa long story. Meanwhile mathematicians were using spherical geometry all thetime, a geometry which obeys the other axioms of Euclidean Geometry andcontains many of the same theorems, but in which the parallel postulate does nothold. All along they had an example of a Non-Euclidean Geometry under theirnoses.Thursday, January 17, 2013
- 7. 7|PageThink of a line L and a point P not on L. The big question is: "How many lines canbe drawn through P parallel to L?" In Euclidean Geometry the answer is ``exactlyone" and this is one version of the parallel postulate. If the sum of the angles ofevery triangle in the geometry is π radians then the parallel postulate holds andvice versa, the two properties are equivalent.In spherical geometry, the basic axioms which we assume (the rules of the game)are different from Euclidean Geometry - this is a Non-Euclidean Geometry. Wehave seen that in spherical geometry the angles of triangles do not always add upto π radians so we would not expect the parallel postulate to hold. In sphericalgeometry, the straight lines (lines of shortest distance or geodesics) are greatcircles and every line in the geometry cuts every other line in two points. Theanswer to the big question about parallels is``If we have a line L and a point P noton L then there are no lines through P parallel to the line L."The Greek mathematicians (for example Ptolemy c 150) computed themeasurements of right angled spherical triangles and worked with formulae ofspherical trigonometry and Arab mathematicians (for example Jabir ibn Aflah c1125 and Nasir ed-din c 1250) extended the work even further. The formuladiscussed in this article was discovered by Harriot in 1603 and published by Girardin 1629. Further ideas of the subject were developed by Saccerhi (1667 - 1733).All this went largely un-noticed by the 19th century discoverers of hyperbolicgeometry, which is another Non-Euclidean Geometry where the parallel postulatedoes not hold. In spherical geometry (also called elliptic geometry) the angles oftriangles add up to more than π radians and in hyperbolic geometry the angles oftriangles add up to less than π radians.Thursday, January 17, 2013

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